Open Games

Open Games Skill (ERGODIC 0)

Compositional game theory via Para/Optic structure

Trit: 0 (ERGODIC)
Color: #26D826 (Green)
Role: Coordinator/Transporter

Core Concept

Open games are morphisms in a symmetric monoidal category:

        ┌───────────┐
   X ──→│           │──→ Y
        │  Game G   │
   R ←──│           │←── S
        └───────────┘

Where:

• X → Y: Forward play (strategies)
• S → R: Backward coplay (utilities)

The Para/Optic Structure

Para Morphism

Para p a b = ∃m. (m, p m a → b)
-- Existential parameter with action

Optic (Lens Generalization)

Optic p s t a b = ∀f. p a (f a b) → p s (f s t)
-- Profunctor optic for bidirectional data

Open Game as Optic

OpenGame s t a b = 
  { play    : s → a
  , coplay  : s → b → t
  , equilibrium : s → Prop
  }

Composition

Sequential (;)

G ; H = Game where
  play = H.play ∘ G.play
  coplay = G.coplay ∘ (id × H.coplay)

Parallel (⊗)

G ⊗ H = Game where
  play = G.play × H.play
  coplay = G.coplay × H.coplay

Nash Equilibrium via Fixed Points

isEquilibrium :: OpenGame s t a b → s → Bool
isEquilibrium g s = 
  let a = play g s
      bestResponse = argmax (\a' → utility (coplay g s (respond a')))
  in a == bestResponse

Compositional Equilibrium

eq(G ; H) = eq(G) ∧ eq(H)  -- under compatibility

Integration with Unworld

(defn opengame-derive 
  "Transport game through derivation chain"
  [game derivation]
  (let [; Forward: strategies through derivation
        forward (compose (:play game) (:forward derivation))
        ; Backward: utilities through co-derivation  
        backward (compose (:coplay game) (:backward derivation))]
    {:play forward
     :coplay backward
     :equilibrium (transported-equilibrium game derivation)}))

GF(3) Triads

temporal-coalgebra (-1) ⊗ open-games (0) ⊗ free-monad-gen (+1) = 0 ✓
three-match (-1) ⊗ open-games (0) ⊗ operad-compose (+1) = 0 ✓
sheaf-cohomology (-1) ⊗ open-games (0) ⊗ topos-generate (+1) = 0 ✓

Commands

# Compose games sequentially
just opengame-seq G H

# Compose games in parallel
just opengame-par G H

# Check Nash equilibrium
just opengame-nash game strategy

# Transport through derivation
just opengame-derive game deriv

Economic Examples

Prisoner's Dilemma

prisonersDilemma :: OpenGame () () (Bool, Bool) (Int, Int)
prisonersDilemma = Game {
  play = \() → (Defect, Defect),  -- Nash
  coplay = \() (p1, p2) → payoffMatrix p1 p2
}

Market Game

market :: OpenGame Price Price Quantity Quantity
market = supplyGame ⊗ demandGame
  where equilibrium = supplyGame.eq ∧ demandGame.eq

Categorical Semantics

OpenGame ≃ Para(Lens) ≃ Optic(→, ×)

Composition: 
  (A ⊸ B) ⊗ (B ⊸ C) → (A ⊸ C)  -- via cut
  
Tensor:
  (A ⊸ B) ⊗ (C ⊸ D) → (A ⊗ C ⊸ B ⊗ D)

References

• Ghani, Hedges, et al. "Compositional Game Theory"
• Capucci & Gavranović, "Actegories for Open Games"
• Riley, "Categories of Optics"
• CyberCat Institute tutorials

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